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scalation.math

Combinatorics

object Combinatorics extends Error

The Combinatorics object provides several common combinatorics functions, such as factorial permutations, combinations, gamma and beta functions.

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  5. def betaF(a: Double, b: Double): Double

    Compute the beta function 'B(a, b)' for the following two cases: (1) when 'a' or 'b' are integers and (2) when 'a' or 'b' are integers + 1/2.

    Compute the beta function 'B(a, b)' for the following two cases: (1) when 'a' or 'b' are integers and (2) when 'a' or 'b' are integers + 1/2.

    a

    the first parameter, a real number satisfying (1) or (2)

    b

    the second parameter, a real number satisfying (1) or (2)

    See also

    http://mathworld.wolfram.com/BetaFunction.html

  6. def choose(n: Int, k: Int, l: Int): Long

    Compute trinomial coefficients: 'n' choose '(k, l'), combinations of 'n' things, '(k, l)' at a time, using Pascal's Tetrahedron.

    Compute trinomial coefficients: 'n' choose '(k, l'), combinations of 'n' things, '(k, l)' at a time, using Pascal's Tetrahedron. Ex: Given 'n' balls, counts ways in which 'k' are chosen for group 1 and 'l' are chosen for group 2.

    n

    the total number of items

    k

    the of items to choose

    l

    the of items to choose (requires 0 <= k + l <= n)

    See also

    http://people.sju.edu/~pklingsb/bintrin.pdf

  7. def choose(n: Int, k: Int): Long

    Compute binomial coefficients: 'n' choose 'k', combinations of 'n' things, 'k' at a time, using Pascal's Triangle.

    Compute binomial coefficients: 'n' choose 'k', combinations of 'n' things, 'k' at a time, using Pascal's Triangle.

    n

    the total number of items

    k

    the of items to choose (requires k <= n)

    See also

    http://www.mathsisfun.com/pascals-triangle.html

  8. def chose(n: Int, k: Int): Long

    Compute 'n' choose 'k' (combinations of 'n' things, 'k' at a time).

    Compute 'n' choose 'k' (combinations of 'n' things, 'k' at a time). A more efficient implementation is given below.

    n

    the total number of items

    k

    the of items to choose (requires k <= n)

  9. def clone(): AnyRef
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  10. final def eq(arg0: AnyRef): Boolean
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  11. def equals(arg0: AnyRef): Boolean
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  12. def fac(k: Int): Double

    Compute 'k' factorial 'k!' using three techniques (requires 'k <= 170').

    Compute 'k' factorial 'k!' using three techniques (requires 'k <= 170').

    k

    the nonnegative integer-valued argument to the factorial function

  13. final def flaw(method: String, message: String): Unit

    Show the flaw by printing the error message.

    Show the flaw by printing the error message.

    method

    the method where the error occurred

    message

    the error message

    Definition Classes
    Error
  14. def gammaF(a: Double): Double

    Compute the gamma function 'gamma (a)' using the Lanczos Approximation.

    Compute the gamma function 'gamma (a)' using the Lanczos Approximation.

    a

    the parameter, a real number

    See also

    http://en.wikipedia.org/wiki/Lanczos_approximation

  15. final def getClass(): Class[_ <: AnyRef]
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  16. def hashCode(): Int
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  17. def hyp2f1(a: Double, b: Double, c: Double, z: Double): Double

    Compute the Gauss Hypergeometric function '2F1(a, b, c; z)' using a power series expansion.

    Compute the Gauss Hypergeometric function '2F1(a, b, c; z)' using a power series expansion.

    a

    the first parameter, a real/complex number

    b

    the second parameter, a real/complex number

    c

    the third parameter, a real/complex number, may not be a negative integer

    z

    the variable, a real/complex number s.t. |z| < 1

    See also

    http://dx.doi.org/10.1016/j.cpc.2007.11.007

    en.wikipedia.org/wiki/Hypergeometric_function For faster or more robust algorithms,

    people.maths.ox.ac.uk/porterm/research/pearson_final.pdf

  18. def iBetaF(z: Double, a: Double, b: Double): Double

    Compute the incomplete beta function 'B(z; a, b)', a generalization of the beta function 'z = 1'.

    Compute the incomplete beta function 'B(z; a, b)', a generalization of the beta function 'z = 1'.

    z

    the variable, a real/complex number s.t. 0 <= |z| <= 1

    a

    the first parameter, a real/complex number > 0

    b

    the second parameter, a real/complex number > 0

    See also

    http://mathworld.wolfram.com/IncompleteBetaFunction.html

  19. final def isInstanceOf[T0]: Boolean
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    Any
  20. val lfac: Array[Long]

    Table of all factorial numbers that can be represented as a long (64-bit) integer

  21. def logfac(k: Int): Double

    Compute the natural log factorial 'ln (k!)' so 'k! = exp (logfac (k))'.

    Compute the natural log factorial 'ln (k!)' so 'k! = exp (logfac (k))'. The formula is a log transformation of Ramanujan's Factorial Approximation.

    k

    the value to take the log factorial of

  22. def mfac(k: Int): Long

    For small 'k', compute 'k' factorial by iterative multiplication.

    For small 'k', compute 'k' factorial by iterative multiplication.
    k! = k * (k-1) * ... * 2 * 1

    k

    the nonnegative integer-valued argument to the factorial function

  23. def mortici(k: Int): Double

    Compute 'k!' using Mortici's Factorial Approximation (more accurate than Stirling's 2nd Order Factorial Approximation).

    Compute 'k!' using Mortici's Factorial Approximation (more accurate than Stirling's 2nd Order Factorial Approximation).

    k

    the nonnegative integer-valued argument to the factorial function

    See also

    http://but.unitbv.ro/BU2010/Series%20III/BULETIN%20III%20PDF/Mathematics/Mortici.pdf

  24. final def ne(arg0: AnyRef): Boolean
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  25. final def notify(): Unit
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  26. final def notifyAll(): Unit
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  27. val pascalTet: Array[Array[Array[Int]]]

    Initial part of Pascal's Tetrahedron, precomputed to speed calculations (Trinomial Coefficients)

    Initial part of Pascal's Tetrahedron, precomputed to speed calculations (Trinomial Coefficients)

    See also

    https://sites.google.com/site/pascalloids/pascal-s-pyramid-3-var

  28. val pascalTri: Array[Array[Int]]

    Initial part of Pascal's Triangle, precomputed to speed calculations (Binomial Coefficients)

  29. def perm(n: Int, k: Int): Long

    Compute permutations of 'k' items selected from 'n' total items.

    Compute permutations of 'k' items selected from 'n' total items.

    n

    the total number of items

    k

    the of items selected

  30. def rBetaC(z: Double, a: Double, b: Double): Double

    Compute the complement of the regularized (incomplete) beta function '1.0 - I(z; a, b) = I(1.0 - z; b, a)'.

    Compute the complement of the regularized (incomplete) beta function '1.0 - I(z; a, b) = I(1.0 - z; b, a)'.

    z

    the variable, a real/complex number s.t. 0 <= |z| <= 1

    a

    the first parameter, a real/complex number > 0

    b

    the second parameter, a real/complex number > 0

  31. def rBetaF(z: Double, a: Double, b: Double): Double

    Compute the regularized (incomplete) beta function 'I(z; a, b)'.

    Compute the regularized (incomplete) beta function 'I(z; a, b)'.

    z

    the variable, a real/complex number s.t. 0 <= |z| <= 1

    a

    the first parameter, a real/complex number > 0

    b

    the second parameter, a real/complex number > 0

    See also

    http://mathworld.wolfram.com/RegularizedBetaFunction.html

  32. def ramanujan(k: Int): Double

    Compute 'k!' using Ramanujan's Factorial Approximation (more accurate than Mortici's Factorial Approximation).

    Compute 'k!' using Ramanujan's Factorial Approximation (more accurate than Mortici's Factorial Approximation).

    k

    the nonnegative integer-valued argument to the factorial function

    See also

    http://files.ele-math.com/articles/jmi-05-53.pdf

  33. def rfac(k: Int, x: Double = 1.0): Double

    Compute the 'k'th degree rising factorial of 'x'.

    Compute the 'k'th degree rising factorial of 'x'. When 'x = 1', this is the regular factorial function 'k!'. Also known as Pochhammer's symbol. Caveat: only works when 'k' is a nonnegative integer

    k

    the number of factors in the product

    x

    the base number to start the product

  34. def stirling(k: Int): Double

    Compute 'k!' using Stirling's 2-nd Order Factorial Approximation.

    Compute 'k!' using Stirling's 2-nd Order Factorial Approximation.

    k

    the nonnegative integer-valued argument to the factorial function

    See also

    http://en.wikipedia.org/wiki/Stirling%27s_approximation

  35. final def synchronized[T0](arg0: => T0): T0
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  37. final def wait(arg0: Long, arg1: Int): Unit
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  38. final def wait(arg0: Long): Unit
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  39. final def wait(): Unit
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